• Sonuç bulunamadı

∑ İ D UZAYINDA ÇEMBER İ Ç İ N HOLDITCH TEOREM İ 2-BOYUTLU ÖKL ** Gülay KORU YUCEKAYA*, H. Hilmi HACISAL İ HO Ğ LU** HOLDITCH’S THEOREM FOR CIRCLES IN 2-DIMENSIONAL EUCLIDEAN SPACE

N/A
N/A
Protected

Academic year: 2021

Share "∑ İ D UZAYINDA ÇEMBER İ Ç İ N HOLDITCH TEOREM İ 2-BOYUTLU ÖKL ** Gülay KORU YUCEKAYA*, H. Hilmi HACISAL İ HO Ğ LU** HOLDITCH’S THEOREM FOR CIRCLES IN 2-DIMENSIONAL EUCLIDEAN SPACE"

Copied!
6
0
0

Yükleniyor.... (view fulltext now)

Tam metin

(1)

G. Koru Yücekaya, H.H. Hacısalihoğlu

)

,

HOLDITCH’S THEOREM FOR CIRCLES IN 2-DIMENSIONAL EUCLIDEAN SPACE

Gülay KORU YUCEKAYA*, H. Hilmi HACISALİHOĞLU**

* Ahi Evran University, Faculty of Art and Sciences, Department of Mathematics, 40200, Kırşehir, TURKEY, gkoruyucekaya@yahoo.com.tr

**

Ankara University, Faculty of Sciences, Department of Mathematics, Tandoğan, 06100, Ankara, TURKEY, hacisali@science.ankara.edu.tr

Geliş Tarihi: 29.12.2008 Kabul Tarihi: 09.03.2009 ABSTRACT

The present study expresses and proves Holditch’s Theorem for two different circles in two-dimensional Euclidean space through a new method.

Key Words: Affine space, Euclidean space, Euclidean circle, Holditch’s Theorem

2-BOYUTLU ÖKLİD UZAYINDA ÇEMBER İÇİN HOLDITCH TEOREMİ

ÖZET

Bu çalışmada, 2-boyutlu Öklid uzayında farklı iki çember için Holditch Teoremi yeni bir metodla ifade ve ispat edilmiştir.

Anahtar Kelimeler: Afin Uzay, Öklid uzayı, Öklid çemberi, Holditch Teoremi

1. INTRODUCTION

In this section we give basic definitions and theorems used in this study.

Definition 1.1 Let A be a non-empty set and V be a vector space on a field F.

i. For ∀P Q R A, , ∈ , f P Q

(

,

)

+ f Q R

(

,

)

= f P R

(

.

(

,

)

f P Q =α α V

∀ ∈ , there is a unique point Q A

ii. For ∀ ∈P A and so that .

If there is a function f AxA: →V, satisfying above propositions, then A is called an Affine space associated to V [1].

Definition 1.2 Let A be a real affine space and let V be a vector space associated to A. Using Euclidean inner product operation on V,

(

1

) (

1

1

, , ,..., , ,...,

n

i i n n

)

X Y x y X x x Y y y

→ →

=

= =

, :VxVIR,

(2)

we can define the metric concepts such as distance and angle in A. Therefore, Affine space A is called a in Euclidean space and is denoted by A E= n [1].

:IRnIR+ Definition 1.3 The transformation defined by

,

X X X

→ →

=

X is called the norm of the vector .

X

Definition 1.4 For X Y→ →, ∈IRn , the measure of the angle between and is the real number Y θ derived from

, cos

X Y X Y θ

→ →

= .

, ,

AB =c AC =b BC a ABCΔ

Definition 1.5 (The Pythagoras Theorem) In a right triangle called , if =

2

, then a2 =b2+c (Figure 1.1) [2].

A

Figure 1.1 Right triangle ABCΔ

C c

a

b

B

Definition 1.6 The set of the points in which are at a distance r from a point M is called a circle with center M and radial length r Euclid circle and is denoted by

E2

: , consta

C=⎧⎨X MX =r r= ⎫⎬

⎩ nt [2]. ⎭

Definition 1.7 The area of a circle with radius r and with point A on it is ( A) 2

A Cr [2].

Theorem 1.1 (Holditch’s Theorem) Let a chord with constant lengths of a b+ of a closed convex curve α be divided by a point on it into two segments with and b as lengths. Let us move the end-points of the chord so that they will entirely trace the curve. Then, the difference between the sizes of the area bounded by the closed curve drawn by point P and that bounded by the main convex curve

a P

πab is [3].

α

(3)

G. Koru Yücekaya, H.H. Hacısalihoğlu 2. THE CLASSICAL HOLDITCH THEOREM

Theorem 2.1 Let an AB chord with a constant length of a b+ on a circle (C) with a radius r in Euclidean plane be divided by a point D into two segments with lengths of a and b, respectively. When the end-points A and B of the chord draw the circle in full, then geometric location of D forms an inner circle (Figure 2.1).

E2

(C)

A

M

O

B a

.

b D

Figure 2.1 Geometric location of D on the chord 2 constant

AB = + =a b l=

Proof. . Let M be the midpoint of AB. Then,

2 . AD a

BD b

MA MB a b l

=

=

= = + =

From the right triangle OMΔA

2 2

OA = OM + MA2

or

2 2

OM = OAMA2

2 2

2 ra b+ ⎞

= − ⎜ ⎟

⎝ ⎠

MD = − b l

2 b a b+

= −

2 b a− = .

Similarly, from the right triangle OMDΔ

(4)

2 2

OD = OM + MD2

2 2

2

2 2

a b b a

r ⎛ + ⎞ ⎛ −

= −⎜⎝ ⎟⎠ +⎜⎝

⎞⎟

r ab

= −

2 .

Since a, b and r are constant, OD is also constant. Therefore, the geometric location of D is a circle with a centre O and a radial length of r2ab.

Theorem 2.2 Let a chord AB with a constant length of a b+ on a circle (C) with a radius R in Euclidean plane be divided by a point D into two segments with lengths of a and b, respectively. When the end-points A and B of the chord draw the circle in full, the size of the ring-shaped region between the orbit of D (inner circle) and circle (C) is independent from the radial length of circle (C).

E2

z= r2ab

Proof. For the radial length of the inner circle is and its two chords AB and EF intersecting at point D of circle (C), let the chord EF be the diameter of (C). The chords AB and EF are divided by D into line segments whose lengths are a, b and r+z, r-z, respectively (Figure 2.2).

A

B F

E r

z

r-z

r+z

r

(C) O

D

a

b

Figure 2.2 Two chords intersecting in circle (C)

Since the triangles BDEΔ and FDAΔ in the interior of circle (C) are similar triangles, BD DE

FD = DA b r r z a

= +

z

2 2

ab r= −z

Since the area of circle (C) is A C( )=πr2 and the area of the inner circle is πz2, the area of the ring-shaped region between these two circles is found as

( )

2 2 2 2

r z r z

π −π =π −

πab

=

(5)

G. Koru Yücekaya, H.H. Hacısalihoğlu

Therefore, the size of the ring-shaped region that falls between the orbit of D and circle (C) is independent from the radial length of circle (C).

Corollary 2.1 The size of the ring-shaped region that falls between the orbit of D and circle (C) is dependent on the selection of point D on the chord; that is, on the segments with lengths of a and b.

3. HOLDITCH’S THEOREM FOR TWO DIFFERENT CIRCLES IN A 2-DIMENSIONAL EUCLIDEAN SPACE

Theorem 3.1 For a circle C with a radial length of R a b+ + and a circle C′ with radius , let an AB rod with a constant length of with end B attached to circle C, and the other end A attached to circle by a joint, be divided by point X on it into two segments with lengths of a and b, respectively. When the rod with the constant length of draws the circles C and C

( )

R< R a b+ + constant ,

a b+ = C′

constant ′

a b+ = with its end-points within these

circles C and C , the geometric location of X forms another inner circle (Figure 3.1). During this motion, the relation between the regions bounded by circles is independent from the selection of circles C and C .

A X

B

O a b

R

C′

C

.

.

Figure 3.1 Circles in E2 Proof.

AX =a, XB =b, OX = +R a

The area of circle C with a radial length R and with a point A on it is ′ ( A) 2

A CR

The area of circle C with a radial length R a b+ + and with a point B on it is

( )

2

( B)

A CR a+ + b

Therefore, the area of the circle, which is the geometric location of point X is

(6)

( )

X

( )

2

A CR+ a

)

Thus,

( ) (

2 2

( B) ( X)

A CA CR a b+ + − R a+ =πb2

(

R a+

)

+b

( )

2

( A) ( X) 2

A CA CRR a+ ⎦ = −πa R a

(

2 +

)

and by adding these two equations side by side, we get

( )

2 2

( )

2

( A) ( B) 2 ( X) 2

A C +A CA CR a b b+ + +RR a+

2 2 2 2

2Rb 2ab b R R 2aR a

π⎡ ⎤

= ⎣ + + + − − − ⎦

( )

2 2

2R b a 2ab b a

π⎡ ⎤

= ⎣ − + + − ⎦

( )( )

2ab b a b a 2R π

= ⎡⎣ + − + + ⎤⎦

( ) ( )

( A) ( B) 2 ( X) 2 2 A CA

A C A C A C π ab b a b a

π

⎡ ⎛ ⎞⎤

⎢ ⎜ ⎟⎥

+ − = + − + +

⎜ ⎟

⎢ ⎝ ⎠⎥

⎣ ⎦

Corollary 3.1 The relation between the regions bounded by circles with radial lengths of and is a and b; in other words, it is independent from the rod’s motion on the circles.

R a b+ + ,

R R a+

REFERENCES

[1] Hacısalihoğlu, H. H., “Diferential Geometry 1st. Ed.”, İnönü University Faculty of Arts and Science Publications, Mat. No.2, Malatya,Turkey, p 895 (1983).(Turkish)

[2] Hacısalihoğlu, H. H., “Analytic Geometry in 2 and 3 dimensional Spaces 2nd. ed.”, Gazi University Faculty of Arts and Science Publications, No.6, Ankara, p 528, (1984).(Turkish)

[3] Holditch, H., “Geometrical Theorem”, The Quarterly Journal Of Pure And Applied Math., Vol. II, p. 38, London, (1858).

Referanslar

Benzer Belgeler

Notice that because the illustrative problem is a multiple objective linear program, the optimization problems ( Pd) and ( Pr,A)with ~ = e that are solved during the global

Sonuç olarak bitkilerin sahip oldukları ağaç, ağaççık, çalı ve yerörtücü gibi boyutsal özellikleri, ölçü, renk, doku ve form gibi tasarım elemanı

成)。 十六、利用紫外線照射進行青春痘粉刺的護理有何功效?

Son adet tarihine göre belirlenen klinik gebelik haf- tas› ile CRL’ye göre ultrasonografik gebelik haftas› aras›ndaki gün olarak ifade edilen farka bak›ld›¤›nda genel

This paper addresses these emerging expectations of a more visible EU role in conflict management by looking at the impact of completed/ongoing EU – CSDP operations in the

Sunulan bu tez çalışmasında 6-8 aylık yaştaki erkek farelerde (Swiss Albino) 60 gün süreli olarak % 40 ve % 60 oranların- da yem (kalori) kısıtlaması uygulanmış; kan

Aim: The present study was carried out to determine some morphological traits of Tarsus Çatalburun breed of Turkish hunting dogs under breeding condition in their homesteads,

Tablo 3’ten de görüleceği gibi, kayıp yaşayan kişinin yaşı, eğitim düzeyi, kaybın ardından geçen süre ve kaybedilen kişinin yaşı kontrol edildikten sonra YAYYE